Question: Solve for $x$ : $ 5|x - 8| + 4 = 1|x - 8| + 9 $
Solution: Subtract $ {1|x - 8|} $ from both sides: $ \begin{eqnarray} 5|x - 8| + 4 &=& 1|x - 8| + 9 \\ \\ { - 1|x - 8|} && { - 1|x - 8|} \\ \\ 4|x - 8| + 4 &=& 9 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 4|x - 8| + 4 &=& 9 \\ \\ { - 4} &=& { - 4} \\ \\ 4|x - 8| &=& 5 \end{eqnarray} $ Divide both sides by ${4}$ $ \dfrac{4|x - 8|} {{4}} = \dfrac{5} {{4}} $ Simplify: $ |x - 8| = \dfrac{5}{4}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 8 = -\dfrac{5}{4} $ or $ x - 8 = \dfrac{5}{4} $ Solve for the solution where $x - 8$ is negative: $ x - 8 = -\dfrac{5}{4} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& -\dfrac{5}{4} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& -\dfrac{5}{4} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $4$ $ x = - \dfrac{5}{4} {+ \dfrac{32}{4}} $ $ x = \dfrac{27}{4} $ Then calculate the solution where $x - 8$ is positive: $ x - 8 = \dfrac{5}{4} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& \dfrac{5}{4} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& \dfrac{5}{4} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $4$ $ x = \dfrac{5}{4} {+ \dfrac{32}{4}} $ $ x = \dfrac{37}{4} $ Thus, the correct answer is $x = \dfrac{27}{4} $ or $x = \dfrac{37}{4} $.